A New Theory for the Buckling of Thin Cylinders Under Axial

A New Theory for the Buckling of Thin
Cylinders Under Axial Compression
and Bending
The resu lts o f exp erim en ts o n axial lo a d in g o f cylin d rical
sh ells (th in en o u g h to bu ck le b elow th e e la stic lim it an d
to o short to b u ck le as E uler co lu m n s) are n o t in good agree­
m e n t w ith previous th eories, w h ich have b een based o n th e
a ssu m p tio n s o f perfect in itia l sh ap e an d in fin ite sim a l d e­
flections. E xperim en tal fa ilu re stresses ran ge fro m 0.6
to 0.15 o f th e th eo retica l. T h e d iscrep an cy is a p p a ren tly
considerably greater for brass an d m ild -s te e l sp ecim en s
th a n for d u ra lu m in an d in creases w ith th e r a d iu s-th ic k n ess ratio. T here is a n eq u a lly great d iscrep an cy b etw een
observed an d predicted sh ap es o f b u ck lin g d eflection s.
In th is paper a n ap proxim ate la rg e-d eflectio n th eo ry is
developed, w h ich p erm its in itia l e c c e n tr icitie s or d ev ia ­
tio n s from cylin d rical shap e to be con sid ered . T rue in ­
sta b ility is, o f course, im p o ssib le u n d er su ch c o n d itio n s;
th e stress d istrib u tio n is n o lo n g er u n ifo rm , an d it is
assu m ed th a t final failure ta k es place w h en th e m a x im u m
stress reaches th e y ield p o in t. T h e effect o f in itia l ec c en ­
tricities an d o f large d eflections is m u c h greater th a n for
th e case o f sim p le stru ts. M ea su rem en ts o f in itia l e c ­
cen tricities in a ctu a l cylind ers have n o t b een m a d e; h o w ­
ever, it is sh ow n th a t m o st o f th e se d iscrep an cies c a n be
explained i f th e in itia l d ev ia tio n s fro m cy lin d rica l fo rm are
assu m ed to be resolved in to a d o u b le h a rm o n ic series, an d
i f certa in reason ab le a ssu m p tio n s are m a d e as to th e m a g ­
n itu d e s o f th e se co m p o n e n ts o f th e d ev ia tio n s. W ith
th e s e a ssu m p tio n s th e fa ilin g stress is fo u n d to b e a fu n c ­
tio n o f th e y ie ld p o in t a s w ell as o f th e m o d u lu s o f e la s tic ­
it y a n d th e r a d iu s-th ick n e ss ra tio . O n th e b asis o f th is
a te n ta tiv e d esig n fo rm u la [5] is proposed, w h ic h involves
re la tio n s su g g ested b y th e th eo ry b u t is b ased o n experi­
m e n ta l d ata.
I t is sh o w n th a t sim ila r d iscrep a n cies b etw een experi­
m e n ts a n d previous th eo ries o n th e b u ck lin g o f th in c y lin ­
ders in p u re b en d in g ca n be rea so n a b ly exp lain ed o n th e
sa m e b asis, a n d th a t th e m a x im u m b en d in g stress ca n
be ta k e n as a b o u t 1.4 tim e s th e va lu es giv en by E q u a tio n
[5]. It is a lso sh o w n th a t p u z z lin g fea tu res in m a n y o th er
b u ck lin g p rob lem s c a n prob ab ly be exp lain ed by sim ila r
co n sid era tio n s, a n d it is h o p ed th a t th is d iscu ssio n m a y
h elp to o p en a n ew field in th e stu d y o f b u ck lin g p rob lem s.
T h e la rg e-d eflectio n th eo ry developed in th e paper sh o u ld
be u se fu l in exp lorin g th is field, a n d m a y be u sed in o th er
a p p lica tio n s a s w ell.
T h e paper p resen ts th e resu lts o f a b o u t a h u n d red new
te s ts o f thin, cy lin d ers in ax ial c o m p ressio n a n d b en d in g ,
w h ich , to g e th er w ith n u m e ro u s te s ts b y L u n d q u ist,3 form
th e ex p erim en ta l evid en ce for th e c o n c lu sio n s arrived a t.
Sym bols U sed
radial displacement under load (w = w2— is the movement due
to the load)
W , W i = numbers proportional to the amplitudes of w and Wi
f = a stress function, defined by Equation [15]
L r, Ls = wave-lengths of the deformation in the x and s
€xj €sj €x8) Kxy KSf Kxs = extensional and flexural strains of the
middle surface of the cylinder wall
T„ Txs, Ox, Gs, Gxs = internal forces and moments per unit
length of section as shown in Fig. 11
a, n = constants having to do with the assumed shape of the
initial displacement, defined by Equation [4]
= the internal strain energy due to the deformation
c = \ / 12(1 — m2) ( ~ 3.3 for engineering metals)
E, n, <r,, = the modulus of elasticity, Poisson’s ratio, and yieldpoint stress of the material
a = average compressive stress in the axial direction, produced
by the external load
r, t = mean radius and wall thickness of cylinder
x, s = axial and circumferential coordinates
u, v, w = axial, circumferential, and radial displacements of
the middle surface of the wall as shown in Fig. 10.
Wi, u>2 = the initial radial displacement considered, and the
1 The experim ental work and m uch of the theoretical work were
carried out in the G uggenheim A eronautical Laboratory of the Cali­
fornia In stitu te of Technology. Presented at the Fourth Interna­
tional Congress for Applied M echanics, Cam bridge, England, 1934.
1 G oodyear-Zeppelin Corporation, Akron, Ohio. M em . A .S .M .E .
Dr. D onnell w as graduated from the m echanical-engineering course
of the U niversity of M ichigan, 1915, and received the degree of P h .D .
in m echanics from the sam e un iversity in 1930. H e had autom o­
bile engineering experience from 1915 to 1922. H e was instructor
and assistant professor of engineering m echanics at the Univer­
sity of M ichigan from 1923 to 1930. H e w as in charge of the struc­
tures laboratory, Guggenheim Aeronautical Laboratory, California
In stitu te of Technology, 1930-1933.
Contributed b y the A eronautics D ivision for presentation at the
Annual M eeting, N ew York, N . Y ., D ecem ber 3 to 7, 1934, of T h e
A m e k ic a n S o c ie t y
e c h a n ic a l
n g in e e r s .
D iscussion of this paper should be addressed to the Secretary,
A .S.M .E ., 29 W est 39th Street, N ew York, N . Y ., and will be accepted
until January 10,1935, for publication in a later issue of Transactions.
N o t e : Statem ents and opinions advanced in papers are to be
understood as individual expressions of their authors, and n ot those
of the Society.
G e n e r a l D is c u s s io n o f P r o b l e m a n d R e s u l t s
HIS paper applies to cylinders thin enough to buckle below
the elastic limit, and too short to buckle first as Euler
columns. Such cylinders, if carefully made and tested,
buckle in small regular waves, as shown in Fig. 1.
s N.A.C.A. R eport No. 473.
I t is shown in Appendix 1 that, if length and end conditions
are neglected, a theory based on the assumption of infinitesimal
deflections and perfect initial shape leads to theoretical values for
the buckling stress under axial load, and the wave-length of the
buckling deflection given by the following:
These results were obtained by Robertson4 by a simplification of
equations obtained by Southwell. The same results can be ob­
tained by neglecting items which experiments show to be negli­
gible in a still more complete solution obtained by Timoshenko.5
In Appendix 1 a much shorter derivation is given, based on simpli­
fied equilibrium equations developed by the author.6 The same
results can also be obtained by energy considerations.
In Fig. 2 this theoretical value of P is compared with the
values of P given by some hundred tests made by the author and
by Lundquist.3 Numerous striking discrepancies will be noted:
(а) There is a great scattering of the experimental points.
(б) All the experimental values of P, and therefore of the
buckling stress <r, are very much lower than the theoretical
F ig . 1
T y p ic a l F a il u r e s o f T h in C y l in d e r s in A x ia l C o m p r e s ­
s io n
, In all the experiments cited in this paper the ends of the cylin­
ders were clamped or fixed in some way. This stabilized the wall
of the cylinder near the ends to such an extent th a t buckling
always started a t some distance from the ends. When cylinders
are tested free-ended, eccentricity of loading and other local condi­
tions at the ends are likely to obtain. Hence, buckling failure
may take place at the ends a t a lower load than would be required
to buckle the main part of the cylinder. Such local effects are
not im portant in most practical applications (as the ends are
usually fastened with some degree of fixity) and will not be con­
sidered in this paper.
As is evident from the illustrations, the buckling waves ob­
served in experiments are comparatively small. In most cases
there were about ten waves around the circumference, and the
wave-length in the axial direction was invariably of about the
same size as in the circumferential direction. This immediately
suggests that the length of the cylinder should have little effect on
its buckling load unless it is very short (with a length less than
one or two wave-lengths). This conclusion is entirely borne out by
the tests. No correlation could be observed between the length
and strength, although many series of cylinders, identical except
for length, were tested. In a great many cases buckling occurred
over only a comparatively small part of the length, the rest of
the cylinder remaining entirely unbuckled. I t is evident from
these facts that, except for very short cylinders, the exact degree
of end fixity (provided it is sufficient to insure against local weak­
ness as already mentioned) can have little effect on the buckling
strength, as different degrees of end fixity are roughly equivalent
to different effective lengths. I t will be assumed in this paper
th a t the cylinders are long enough so th at length and end condi­
tions can be neglected. (The tests indicate th at they can be ne­
glected even when the length-radius ratio is considerably less
than one.)
F ig . 2
C o m p a r is o n o f E x p e r im e n t a l S t r e n g t h s o f T h in C y l in ­
U n d e r A x ia l C o m p r e s s io n W it h C l a s s ic a l T h e o r y
(c) Instead of being constant, the experimental values of P
show a very decided tendency to become smaller with increasing
(d) The experiments made by the author, which were made on
brass and steel specimens, give consistently lower values of P
than those made by Lundquist, which were made with duralumin
As to the shape of the buckling deflections, a first discrepancy is
th a t in all experiments the wave-lengths in the axial and circum­
ferential directions are consistently nearly equal, which would
require th a t X = S, whereas Equation [2] requires no definite
relation between X and S b u t only th a t (X + S )2/ X have a
definite value. Thus [2] would be satisfied, for instance, if
4 R . & M . N o. 1185, B ritish A .R .C ., 1929.
• S. Tim oshenko, “Theory of E lasticity," 1914 (in R ussian), p.
* N .A .C .A . R eport N o. 479.
S = 0, X = 1, in which case [1] and [2] reduce to the “sym­
metrical” theory for the buckling of cylinders under axial com­
If we assume that X = S, as indicated by the tests, [2] gives
Fig. 3 shows this value compared with the values found from the
experiments. I t will be seen that again there is great scattering,
but th at the experimental values of X or S are consistently much
smaller than the theoretical; that is, the experimental wave­
lengths are much larger than the theoretical.
Attempts to explain these discrepancies have not been very
satisfactory. The author believes th a t there is nothing incorrect
in the classical analysis described, but th a t the assumption it
makes th a t there are no initial deviations from true cylindrical
shape (or unevennesses in the physical properties of the material
or other imperfections—for the purposes of our discussion these
can be assumed to be replaced by equivalent deviations from
perfect shape) is not permissible in this case, at least for ordinary
test specimens or ordinary practical applications. Theories neg­
lecting initial inaccuracies, which are, of course, always present
to some extent, give good approximations in other buckling prob­
lems, but these inaccuracies seem to be much more im portant in
this case.
I t is easy to explain why this is so. When a developable sur­
face, such as a flat or a cylindrical surface, is deformed to a nondevelopable surface, there are produced (besides the flexural and
stretching strains due to change of radius, which are considered in
usual theories) strains which might be called “large-deflection
strains,” th at result from stretching and compressing the develop­
able shape into a non-developable one. These are of more im­
portance than is commonly realized. In the bending of a strut a
large-deflection theory is not needed unless the deflections are
of the order of magnitude of the length of the strut, and as such
deflections are of little practical interest, large-deflection theories
have received little attention. B ut the aforementioned “largedeflection strains” in sheets become of importance in general
when the deflections are of the order of magnitude of the thick­
ness of the sheet.
For the very thin cylindrical shells which are under considera­
tion (especially those rolled up from sheets, as were the speci­
mens in the tests, and as is the case in all common applications),
the initial deviations from cylindrical form are already of this
order of magnitude. When the compressive load is applied these
initial displacements are, of course, increased. The stresses due
to the “large-deflection strains” accompanying this movement in­
crease very rapidly, and combined with the direct compressive
stress and the other stresses commonly considered, reach the
yield point of the material at certain points long before the load
has risen to the value given by Equation [1 ]. Beyond this point
it is evident th at the resistance of the cylinder will rapidly fall,
so th at complete failure must take place soon afterward.
I t may be argued that the same effect must take place in the
buckling of flat panels, and that, in this case, the ultimate
failure is far above th at given by the usual stability theory. This
is true, but it does not invalidate the explanation given. There
is nothing surprising in the fact that in one case the ultimate
load—reached soon after the most highly stressed material passes
the yield point—comes well above the classical stability limit, while
in the other case it comes well below. In the case of a thin flat
panel the stability limit itself is very low; the load at which
the combination of common and “large-deflection stresses”
reaches the yield point (at the edges of the panel) is much higher,
and the panel can go through the stability limit without complete
7 S. Timoshenko, Zeitschrift filr Mathematik und Physik, vol. 58,
failure, because of the artificial support given the edges. In the
case of a cylinder the classical stability limit is comparatively
very high, and the large-deflection stresses increase very fast ow­
ing to the small size of the waves, so that failure is brought on by
yielding a t a lower load than is indicated by the classical theory.
B ut the two cases are similar in principle, and a complete solution
of the problem of the ultimate strength of flat panels can only
be obtained by using large-deflection theory.8 The simplified
F ig . 3
C o m p a r is o n o f S iz e o f B u c k l in g W a v e s
T e s t s a n d b y C l a s s ic a l T h e o r y
G iv e n
large-deflection theory developed in this paper should be useful
in making such a study. The initial displacements are probably
not important for this case, so th a t it should be possible to obtain
a more definite result than in the problem of the present paper.
In other buckling problems, such as the buckling of struts and
the buckling of thin cylinders under torsion, initial displacements
or other inaccuracies must also be present, and yet in these cases
a reasonable check with experiments is obtained without con­
sidering these questions. The explanation is th a t in these cases,
as in the case of the flat panel, the classical stability limit is below
the load a t which the most highly stressed point would pass the
yield point (because the large-deflection stresses are absent in the
case of a strut, and they are less important in the case of torsion
of a cylinder, as the buckling shape is more nearly a developable
surface) and there is no artificial support which can carry some
part of the structure through the general stability limit without
failure, as is the case with the edges of flat panels. But there
may be unexplored ranges of dimensions or materials where these
questions are important, even in these cases. The author has
noticed th a t in the buckling under torsion of very short cylinders,
for which the buckling shape is further from a developable sur­
face than it is for a long cylinder, the yield point is usually reached
a t about the theoretical stability limit. In this case the stiffen­
ing effect of the large-deflection stresses (and in general such
stresses must always have a stiffening effect, because the increase
in internal energy due to them must be supplied by an increase
in the external load) just about balances the opposite effect of the
yielding of the material, so th a t yielding and deepening of the
buckling deflections continue for a long time with little change in
A v ery approxim ate solution for this problem has been given by
von KArm&n. See “T he Strength of T hin P lates in Com pression,”
A .S .M .E . Trans., vol. 54, 1932, paper A PM -54-5.
This probably explains another puzzling experimental fact,
the external load.9 We evidently have here a third type of prob­
lem in which these questions are important. I t is thought th at
noted by Lundquist,10 which the author corroborates, th at local
these few remarks, and the discussion of the case of the cylinder buckling frequently occurs somewhere in the cylinder without pre­
cipitating complete failure, as the load can continue to increase
under axial compression, may help to clarify other points in the
study of buckling problems, which hitherto have been puzzling.
without any general buckling until complete failure takes place
suddenly over a large part of the cylinder at the load which
Let us now consider in more detail just what happens when a
compressive load is applied to a thin sheet or strut having initial would be expected if no preliminary buckling had taken place.
I t now seems evident th a t final failure will, in general, take
deviations from straightness in the direction of the compression.
Consider first the simple case of a hinged-end strut of a certain place when the stresses due to one of the components of the initial
size. If it has an initial displacement in the shape of a half sine displacement (combined with the direct stress) reach the yield
wave, the amplitude of the displacement will be increased when point. The component which produces final failure (and which
a compressive load is applied. The ratio of increase will be small evidently has the same wave-lengths as the final failure, Fig. 1)
for small loads, but will become very great when the load ap­ is thus the one which causes yielding to take place at the lowest
proaches the theoretical stability limit. If the initial displace­ value of the external load. We can determine the characteristics
of this component from this fact by using minimum principles
ment were in the shape of a full sine wave, the amplitude of this
displacement would also increase when a compressive load is ap­ or their equivalent.
We would naturally expect th at the component having wave­
plied. The ratio of increase would also be small for small loads
and would increase as the load increases. I t would become in­ lengths as given by Equation [2] or [3] would cause this yielding,
finite if the load could be raised to four times the theoretical sta­ and hence final failure, sooner than any other component, as
bility limit. Now imagine that the initial displacement is a com­ its amplitude will certainly increase faster than th a t of any other
bination of the two displacements mentioned. If the displace­ component, as the load is increased. But, as noted before, the
ment is small, the action which takes place will be a superposition tests indicate th at the component actually causing failure always
has a considerably longer wave-length than this. A reasonable
of the two actions just described. The rates of increase of the
explanation for this is th at the initial amplitudes of the different
two components will be enormously different when the load is near
components are not equal. It is certainly natural to suppose that
the theoretical stability limit, but they will not be so very different
the components of the initial radial displacements which have
when the load is small.
Similar actions will take place in the case of a cylinder under longer wave-lengths will, in general, have larger amplitudes as
well. Hence, failure may be precipitated by a component with
axial compression. For convenience in discussing this question
a longer wave-length than [3], rather than by the component with
we may consider the total initial radial displacement from true
cylindrical form to be made up of numerous component displace­ this wave-length, because the first has a much greater initial
amplitude, in spite of the fact th a t the amplitude of the second
ments, each of wave form (similar to the displacements shown in
Fig. 1), but having different wave-lengths in the x, or s, or both increases faster, and in spite of the fact, also, th at the stresses
directions. Now suppose th a t only one of these components of produced by the deformation decrease as the wave-length in­
the displacement is present. As the compressive load is in­ creases. The calculations given later in this paper show that
this is easily possible. In this connection it should be remem­
creased, the amplitude of this component will increase until, at
some point of the wave, because of the combination of the direct bered th a t failure takes place at only a fraction of the theoretical
stress (the average compressive stress in the axial direction re­ stability limit given by [1 ], at which time the component with
sulting from the load) with the other stresses produced by the wave-length [3] does not increase so very much faster than its
deformation, the yield point of the material is reached. Such rival components.
We have been speaking of “components” of the initial displace­
yielding will occur simultaneously at corresponding points in
ment of the cylinder without being very definite as to their shape.
each of the waves of the displacement all over the cylinder.
Actually this is, of course, as stated in the beginning, only a con­
When the load is increased beyond this point, it is obvious th at
venient concept for the purposes of the previous discussion. To
the resistance to this form of displacement will be greatly lowered,
put the question on a definite basis, actual measurements should
and the displacement will increase very rapidly (with probably a
be made of the initial displacements in many specimens. The
falling resistance to the load, which would explain the sudden,
explosive failure which always takes place in tests) until we have author has not had the opportunity of making such measure­
ments and has no data of this sort from other sources. In the
such a condition as is shown in Fig. 1.
Now let us suppose th a t all the components of the displacement absence of such data the only thing th at can be done is to dis­
are present. When a compressive load is applied to the cylinder cover if there is a possibility of explaining the experimental
facts on the basis of reasonable assumptions as to the initial dis­
the amplitudes of all the components will tend to increase.
Stresses will be produced due to each one of these components.
The initial displacement, whatever it may be, can be analyzed
(It is shown later th a t the components can be defined so th a t the
question of interaction between them is of minor importance; into a double harmonic series. When an external compressive
in any case stress systems having the wave patterns of each of the load is applied, all the terms of the series will, in general, tend to
components will be produced—this is all th a t is necessary for the increase in amplitude. But their effects will not be independent,
following argument.) Combination of the stresses produced by as with similar harmonic terms in the case of a strut. I t can be
several different components, together with the direct stress, may said with confidence that, in most cases, the action of one term
cause yielding at certain points in the cylinder before yielding will be nearly independent of the action of another, th at is, the
due to any single component, such as described in the last para­ combined effect will be nearly the same as the sum of their effects
when present separately, but th at there will probably be certain
graph, takes place. But, due to the different wave-lengths of the
different components, such yielding because of combinations of groups of terms whose action will decidedly not be independent.
components will be local. I t will not occur simultaneously at cor­ These will be groups of terms which, taken together, form a shape
which it is “easier” for the cylinder wall to deflect to than a single
responding points in each of the waves of one of the components,
as described before, and hence it will not produce a great weaken­ pure harmonic shape. We would expect such groups to consist of
a primary term, which determines the wave-lengths of the group,
ing in the resistance to any one of the components.
"IV A .C .A . R eport N o. 479, p. 12.
10 N.A.C.A. R eport No. 473, pp. 5 and 14.
are actually present should be sufficient for our purpose—to see
and higher harmonics, which modify the shape of the primary
if the required magnitudes are reasonable. I t might be empha­
term to an “easier” shape; or combinations of such terms with a
secondary term or terms which enable some of the large-deflection sized here th a t it is not expected th a t the assumptions made
stresses to be annulled by stresses due to change of radius. (We as to the nature of the initial displacements are anywhere near
exact. I t is only necessary, for our contention", th a t they repre­
shall use such a term in the calculations given later.)
sent average tendencies—the great observed scattering in the
Each of the “components” of the initial displacement in the
experimental results explains the wide deviations from the as­
previous discussion can be considered to represent one of these
groups of harmonic terms. Since we try to include in each group sumptions which must be expected.
In setting up the theory, a combination of the equilibrium and
all the terms whose actions greatly affect each other, and since
it has been shown previously th a t it is not important to con­ energy principles is used. Expressions for the extensional and
flexural strains of the middle surface of the wall are first set up
sider combinations of the stresses due to different groups, it seems
that we shall make no very great error if we consider only one of by adding terms describing the large-deflection strains to the
the groups and neglect the effect of all the others. The one we usual expressions. Using the usual relations between the internal
forces and the strains, the equilibrium equations of the elements
consider will of course be the one which precipitates failure,
of the cylinder wall in the axial and circumferential direction are
chosen by the condition th at it gives the lowest final failure load.
Any conclusions which we draw from this calculation will be on set up, the same as in small-deflection theory. These enable a
“stress function” to be used, and it is then possible to derive,
the conservative side, because the chief contention to be made is
that the low failure load found in experiments can be explained first, a relation between the stress function and the radial dis­
by reasonable initial displacements, and any parts of the initial placement; and second, an expression for the internal elastic
energy in terms of these two variables and the properties of the
displacement which we neglect could hardly have had any other
cylinder. If, now, we assume an expression for the radial dis­
effect than to reduce the calculated failure load.
placement, we can obtain the corresponding expression for the
In the calculations, which are given in detail in Appendix 2,
it is assumed that the initial displacement and the final displace­ stress function from the first relation, and with the aid of the
ment, and hence also the movement, are of the same geometrical second expression and the principle of virtual work we can ob­
shape. The assumed shape of displacement consists of a pri­ tain the external compressive load required to produce the dis­
placement. Using the expression for the displacement already
mary term taken as a harmonic function of x and s, with the
wave-lengths L x and L a, and the amplitude W \t/c or W t/c (for described, we obtain P (defining the external load) as a function
of X , S, W lt and W.
the initial displacement and the movement), and one secondary
W ith the assumed displacement and the expressions previously
term designed to annul as much as possible of the large-deflection
found for the internal forces, we now set up the condition for yield­
stresses with stresses due to change of radius. Fig. 12a shows the
ing at any point of the displacement wave by the maximum-shearprimary term. It is evident that at p — p the material is on the
average circumferentially stretched, while the material at q — q energy theory. With this expression it would be possible to
is on the average circumferentially compressed, to have equili­ determine the exact point in the wave a t which yielding first
takes place, by maximum-minimum principles. This would
brium in the circumferential direction. By superposing the
symmetrical deformation shown at Fig. 126 of the proper ampli­ be a very complicated calculation, hpweyer, so instead it is as­
tude, we annul this average circumferential stretching and com­ sumed th a t yielding first takes place at the nodes of the waves,
pressing, and although we introduce a certain amount of bending where trial indicates th a t the stress condition is at least approxi­
in the longitudinal direction, the total internal energy is con­ mately as severe as anywhere. W ith this assumption we ob­
' f _ cay r \
siderably reduced, th at is, the addition of the symmetrical term
as a function of X , S, Wi, and W.
tain P,
makes it a much “easier” form. The use of this term simplifies
V \ ~ E ~ t)
the calculations, and trial shows it to be very effective in reducing
If, now, we knew the actual value of Wi, we could eliminate W
the final failure load. Without it, it is impossible to explain at
between the two relations and obtain P as a function of Pv, X , and
all the low values of failure loads found in experiments. I t is, S. We could then, by trial or by using minimum theory, de­
of course, quite probable that a refinement in the magnitude of termine the X and S which make P a minimum (which means
this term, or the addition of other terms, would be found to
determining th a t “component” of the deflection which precipi­
be still more effective in reducing the final failure load. No at­ tates failure, as we previously decided to do), and thus find P
tempt has been made to make such refinements because of the
as a function of P„, which means finding a as a function of the
complexity it would involve.
properties of the cylinder E, r/t, and E /cay.
As to our justification for assuming th a t the initial displacement
As we do not know Wi, we shall do the reverse of this and deter­
and the movement will have the shape assumed, there is no doubt
mine the magnitude of Wi, which, by the above process, gives
th at the movement will take this shape, or a still “easier” one, if values of P as low as shown by the experiments, and then see if
possible. And in neglecting possible refinements in the shape, the
this value of W i is reasonable. However, although we do not
conclusions which we draw from the calculation will be on the
know the absolute magnitude of W i we can say something as to
conservative side for the same reasons as were given in a previous its probable variation—or the probable “average tendencies”
case. The amplitude of the secondary term is proportional to
of its variation—with I, Lx, L,, etc. Thus for the flat sheets from
W 2, th at is, it is a second-order term, and its absolute magnitude,
which cylinders are made we can certainly expect the initial dis­
as given by the calculations, is always very small compared to
placement to decrease with increasing thickness and th a t com­
the primary term, so that its presence would not be noticeable in
ponents of the displacement of shorter wave-length will have
smaller magnitudes. Thus in Fig. 13 we would not expect two
Of course we have no right to assume th a t the two terms are
sheets rolled by the same method, one thin and one thick, to
present in the actual initial displacement with the relative magni­ have components of displacement of like wave-length with the
tudes which we have assumed, although it is possible th a t the
same amplitudes, as a t a and 6, but would rather expect the ampli­
action of curving flat sheet into cylinders tends to bring this about.
tude to be smaller in the thicker sheet, as at c. And in Fig. 14 we
But the effective magnitude of the whole group will be some kind
would not expect components of different wave-length in the same
of average determined by the magnitudes of the terms actu­ sheet to have the same amplitude, as in a and 6, but would expect
ally present, and the assumption th a t the calculated proportions
a smaller amplitude for a shorter wave-length as at c.
F io .
D ia g ra m s
S h o w in g
H o w E x p e r im e n ta l R e s u lt s C a n B e E x p la in e d b y L a r g e - D e f le c tio n T h e o ry i f C e r ta in
s u m p tio n s A r e M a d e a s t o I n i t i a l D e v ia tio n s F ro m C y l i n d r i c a l F o rm
(T h e th eoretical v a lu e X or S cou ld n o t be d eterm in ed v ery e x a c tly w ith o u t a g reat d ea l of labor. V alu es sh o w n are rough estim a tes.)
A s­
It is thus reasonable to expect that for the flat sheets Wi would
vary more or less as given by the equation
where a and n are non-dimensional quantities more or less con­
stant for sheets rolled by the same process.
The process of curving the sheet into cylinders will certainly
change this expression for W i considerably, and doubtless intro­
duce the quantity r into it. The nature of the change is some­
thing which could doubtless be analyzed, but a satisfactory analy­
sis is a difficult problem in itself. An attem pt was made to make
a rough analysis, but the attem pt was abandoned as it was felt
th a t such results were very likely to be misleading. In the ab­
sence of a satisfactory solution, it was felt th a t much could still
be learned by using [4] as it stands, as the argument on which it
is based, already given, certainly holds for curved sheets as well
as flat. Some discussion is given later of possible changes due to
the process of curving the sheet. This question has no effect
on the main contention of this paper, th a t the very low failure
loads found in tests can be explained by reasonably small initial
A principal effect of curving the sheet into cylinders will prob­
ably be to upset the symmetry of expression [4] with respect
to L x and L ,. This will greatly affect the ratio between the values
of L x and L . given by the theory. W ithout some knowledge of
the nature of this effect it is useless to try to see if the theory will
explain the fact th a t Lx and L , are nearly equal in tests. Hence,
the equality of L x and L , (and so of X and S) was assumed, to see
if the other results of tests could be explained.
Using the expressions for P and Py, already mentioned with
[4], we find P as a function of E/cay, r/t, X (or S), n, and a. As
expected, it is found th a t the value of X (or S ) which makes P
a minimum depends on the value of n. I t is found th a t we get
about the value of X shown by tests if n is 5/4, which is certainly
a reasonable value. To bring P down to the level of the test
values, a has to be about 1.1 X 10 ~6. Using these values of n,
X , and a, we obtain P as a function of E/c<ry and r/t.
The value of E/c<ry for the duralumin specimens tested by Lundquist was about 80, while the value of this quantity happened to
F ig . 5
V a l u e s o f W t/c a n d W it/c R e q u i r e d b y L a r g e - D e f l e c tio n T h e o ry t o E x p la in T e s t R e s u lts
be about 165 for both the steel and the brass specimens tested by
the author. For each of these values of E /ca y we obtain P
as a function of r/t. In Fig. 4a the corresponding values of P
and r/t so obtained have been plotted, and the resulting curves
can be compared with the experimental points for the same values
of E /ary. I t will be seen th at the accordance is excellent, the
theoretical curves showing nearly the same downward slope with
increase of r/t, and decrease of P with increasing E /ca y, as shown
by the test results. In Fig. 4b the value of X or S corresponding
to these results is compared with the test results.
Fig. 5 shows the values of Wit/c, and the values of W t/c at which
failure starts, as given by the calculations. The values found for
Wt/c, giving the magnitude of the movement under load up to the
time th at failure starts, are very small, which corresponds to the
test experience th a t no general buckling can be noticed by eye up
to the instant of the sudden failure. The values found for Wit/c,
which indicate the magnitude of the initial displacement which
must be assumed to explain the low test failure loads, do not seem
particularly excessive. They increase with the radius-thickness
ratio, as common observation indicates they should, although this
increase is due to the selection of a component with a longer wave­
length, rather than any effect on the initial displacement by the
curving of the sheet. Of course, these values represent only a
part of the total initial displacement, but probably in practise
quite a large part; the fact that failure can apparently take place
over only a part of the cylinder wall about as easily as over it all,
as indicated by the tests, means that it is not necessary for the
component most favorable to failure to be large or even to be
present all over the cylinder wall, but failure can take place
wherever, by accident, it happens to be large over a considerable
portion of the wall. If measurements finally show th a t initial
displacements are actually not as large as the theory calls for, then
the relative crudity of the calculations is probably to blame. I t
will be remembered that most of the simplifications were on the
conservative side in this respect. Since the crude secondary term
used in the displacement had such an enormous effect in reducing
the calculated values of P for a given value of W i (or reducing the
value of Wi for a given value of P), it is probable th a t further re­
finement would have an important effect in the same direction.
There is not much likelihood th a t the result obtained of check­
ing the experimentally indicated decrease of P with increase of
r/t and of E/c<ry is accidental and due only to the particular as­
sumptions made. The author has tried many combinations and
finds th at any reasonable assumption regarding the initial dis­
placement seems to result in the same general tendency. Thus,
varying the value of n in [4] has little effect on the results except
to change the value of X or S which gives the minimum P. One
effect of curving flat sheets into cylinders is probably to reduce the
size of initial unevennesses, so th at we might expect some power
of r/t in the numerator of [4], after this effect has been allowed
for. Such an addition has also been tried, and it is found that
we still get about the same reduction of P with increase of E/cay,
and, as might be expected, still greater reduction of P with in­
crease of r/t, a greater reduction than is indicated by experi­
ments. However, the unknown effect of bending the sheets on
the relation of Wi to the wave-lengths may easily counterbalance
It is obvious that our assumptions as to the initial displace­
ments, and to a less extent the calculations themselves, are at
best only rough approximations—a fact which may be excused by
the newness and difficulty of the problem. Much work, both
theoretical and experimental, must be done before the question
can be considered as settled. Other factors may enter the prob­
lem besides those which have been discussed. When this paper
was presented recently at the Fourth International Congress of
Applied Mechanics, Prof. R. V. Southwell made a very interesting
suggestion. When a cylinder is compressed, the axial shortening
is accompanied by a circumferential expansion. This expansion
is largely prevented at the ends, where the cylinders are attached
to something else, and this holding-in of the ends relative to
the rest of the cylinder produces (since axial elements of the cylin­
der wall are in the condition of a beam on an elastic foundation)
a symmetrical displacement, similar to that shown at Fig. 126
except th at the amplitude is a maximum at the ends and damps
out as we go toward the center. Calculations show th a t the
wave-length of this displacement is the same as required for the
secondary term of our assumed displacement when X or S is
about 0.13. This value of X or <S is about half th a t given by
the classical stability theory, but is still not as small as the aver­
age of the tests calls for. This phenomenon undoubtedly takes
some part in the buckling action but how important a part it
takes is still to be determined. I t is an experimental fact that
when the buckling occurs over only a part of the length of the
cylinder it usually occurs very near the end, where this displace­
ment is a maximum. (We could not expect the buckling to occur
still closer to the ends, on account of the fixity there.) Some of
the photographs given show this. I t is possible th at this phe­
nomenon may eliminate the necessity for some of the assump­
tions which have been proposed.
In spite of the roughness of the calculations, it is believed that
the most important factors have been taken into consideration
and th a t the results, while they do not prove th a t the discrepan­
cies between the experiments and the classical theory are to be
explained in the general manner indicated, at least make this
strongly probable.
I t is particularly believed th a t it has been demonstrated th at
the dependence of P on r/t and E/cay, indicated by the tests, is
not a mere accident, or explainable by variations in experimental
technique. I t therefore seems th a t this relation should be con­
sidered in design formulas. The most im portant practical signifi-
ig .
o m p a r is o n
q u a t io n
[5 ] W
it h
esu lts
cance of this relation is th a t some improvement in buckling
strength can be allowed for if a material with a higher yield point
is used, and vice versa. The following formula for the average
failure stress
has been found to describe the relations shown by the tests
quite well, and to give reasonable values for the extreme condi­
tions E /o v — 0, E/<rv = °°, and r/t = 0. I t gives a negative re­
sult when r /t is very large; in this case, which is far outside the
practical range, <r can be taken as zero without great error.
Curves obtained from this formula are compared with the test
results in Fig. 6. As stated, the formula is designed to give the
average strengths to be expected, and if it is desired to know the
minimum strength likely to be encountered under any circum­
stances, some factor must be used with it; the value of the factor
to be used under any given conditions can best be estimated from
the test results shown in the figure. In the absence of anything
better, this formula, with or without some factor, is recommended
for design purposes.
No theoretical work has been done on the allied problem of the
buckling of thin cylinders under pure bending. The smallness
of the buckling waves found for axial loading immediately sug­
gested th a t buckling will take place on the compression side of
the bending specimen in practically the same way as it does in
an axially loaded specimen, and th a t any results found for axial
loading would apply also to pure bending, with some factor to
allow for the fact th a t the stress varies from zero to a maximum
instead of being constant around the circumference.
Numerous bending tests on specimens similar to those tested
in axial compression completely confirmed this opinion. Photo­
graphs of typical specimens are shown in Fig. 7, while the results
of the tests are plotted alongside the results obtained in axial
compression, in Fig. 8. Buckling occurred over the compression
side of the specimens in the same wave form, with approximately
the same wave-lengths, as in the axially loaded specimens. It
will be noticed from Fig. 8 th at the results show exactly the
same decrease of P with increase of r/t as shown by the axially
. 7
y p ic a l
F a il u r e s
h in
C y l in d e r s
T h e E x p e r im e n t s
The results of the experiments have already been described.
Detailed data are given in Tables 1 and 2 for the axial compression
and pure bending tests respectively. The specimens were made
in exactly the same way and tested on the same special testing
machine (shown in Fig. 9) as the torsion specimens described in
a previous paper by the author,6 and the reader is referred to this
paper for a detailed description of the testing machine and the
technique of making the specimens. The axially loaded speci­
mens were loaded through a very frictionless universal joint very
carefully centered to insure against eccentricity.
One fact, not previously mentioned, which was noticed in mak­
ing the tests, is th a t no m atter how quickly the loading was
stopped after final failure took place it seemed impossible to
reach as high a load on reloading as was reached the first time—
and this in spite of the fact th a t the average compressive stress
e n d in g
loaded specimens. In computing P the value of a w-as taken as
the maximum stress in bending, according to elementary theory.
The values of P found are about 1.4 times the values found in
axial-compression tests for all values of r/t. This is just about
what would be expected, as it shows th a t general buckling takes
place when the stress at a point in the cylinder wall about 45 deg
to the neutral axis rises to the value which produces failure in a
uniformly stressed specimen.
An ingenious theory for the stability of thin cylinders in bend­
ing has been advanced by Brazier.11 Accordingto this, the elastic
curvature produced in the initially straight cylinder produces
the well-known phenomenon of the flattening of the cross-sections
of curved tubes under bending. The cross-section becomes more
and more oval until a point is reached at which the resistance to
bending starts to decrease, after which, of course, complete col­
lapse takes place. Serious objection can be made to the theoretical
derivation given by Brazier in th a t small-deflection theory is used
and is assumed to apply after the deflections become very large;
th a t is, the small-order terms neglected in the derivation (which
can be neglected when the deflections are very small) are, at the
critical point, of the same magnitude as the terms considered.
However, it is an undoubted fact that this type of failure does
take place in comparatively thick tubes made of a material with
a low modulus, such as rubber tubes and thick metal tubes
stressed above the yield point. I t would seem that Brazier’s
type of failure and the small-wave type are more or less indepen­
dent types of failure, and in an actual tube failure is produced
by whichever type happens to occur first; that is, whichever type
requires the least load. For thin metal tubes of the type con­
sidered in this paper, failure of the small-wave type, the same as
in axial compression, probably always occurs first. For design
purposes it seems safe to say th a t the maximum bending stress
given by the elementary bending formula can rise to about 1.4
times the value given by [5 ] before buckling will, on the average,
. 8
F ig . 9
11 R . & M . N o. 1081, British A .R .C . 1926.
t io n o f
C o m p a r is o n
A x i a i .- C o m p r e s s i o n
T ests
P u r e - B e n d in g
S p e c ia l T e s t in g M a c h in e W h ic h A p p l ie s
T o r s io n , B e n d in g , a n d C o m p r e s s io n (o r T
C o m b in a ­
L oads
e n s io n )
T A B L E 1 A X I A L C O M P R E S S IO N T E S T S '^
due to the load was in many cases
,-------------------- s
—S teel T u b es----------- B rass Tubesonly a small fraction of the yield
T h ick ­
N um T h ick N um ­
point. This tends to bear out
D ia m e­
ness X
F a ilin g ber of
D ia m e ­
n ess
F ailin g ber of
L en gth , 103,
X10-6 load, w a v es in
L en g th , X 103, X io - «
load , w a v es in
the contention that, even for very
( lb /i n .2), (lb) circum .
( lb /in . 2)
(lb) circum .
thin cylinders, the cause of final
5 .0 7
2 .88
3 1 .3
2 38
5 .6 7
5 .8 3
1 6 .3
5 .6 7
2 .7 8
3 1 .3
3 .7 5
5 .9 6
1 6 .3
failure is the fact th at at certain
3 .7 5
2 .9 2
3 1 .3
3 .7 5
5 .9 0
1 6 .3
points, strategically located to
1 .8 8 5
2 .8 4
3 1 .3
5 .6 7
2 .9 8
1 5 .7
weaken the cylinder for the type
1 .8 8 5
2 .7 2
3 1 .3
1 .8 8 5
2 .9 5
1 5 .7
3 1 .3
5 .6 7
2 .1 7
1 .8 8 5
2 .9 6
1 5 .7
of failure involved, the local stress
5 .6 7
3 1 .3
5 .6 7
1 5 .7
3 .7 5
2 .1 8
3 1 .3
5 .6 7
1 5 .7
passes the yield point.
3 .7 5
2 .1 3
3 1 .3
3 .7 5
2 .1 3
1 5 .7
Another short series of tests
1 .8 8 5
2 .0 5
3 1 .3
3 .7 5
2 .0 7
1 5 .7
3 1 .3
1 .8 8 5
2 .0 3
5 .6 9
5 .9 4
which was made, the data for
2 .6 4
5 .6 7
3 1 .3
5 .6 9
5 .8 5
1 6 .3
which are given in Table 3, bears
1 .8 8 5
2 .7 4
3 1 .3
3 .7 5
5 .9 8
1 6 .3
out the same contention. In this
5 .6 7
1 .9 9
3 1 .3
1 .8 8 5
5 .8 7
1 6 .3
3 1 .3
3 .7 5
5 .6 7
1 .9 3
1 5 .7
eight cylinders were tested in axial
1 .8 8 5
2 .8 0
3 1 .3
3 .7 5
2 .0 7
1 5 .7
1 .8 8 5
1 .9 9
3 1 .3
5 .6 7
5 .9 2
6 12
1 6 .3
compression. The cylinders were
3 .7 5
1 6 .3
identical except that instead of
1 .8 8 5
5 .9 7
1 6 .3
being bent to the proper radius be­
fore making the joint, as was done
■ ■
-------Bra i,S8 T ubes—S teel T u b es-----with all the other specimens, some
T h ick ­
N um ­
T h ick ­
N um ­
of them were bent to the proper
F a ilin g ber of
D ia m e ­
F a ilin g
ber of
d iam eter, L en gth , X103
X 1 0 -6 m o m en t w a v es in
L en g th , X108 X 1 0 -6 m o m en t w a v es in
radius, while others were sprung
(lb /in . 2) (in .-lb ) circum .
en .)
( lb /in . 2) (in .-lb ) circum .
into shape from the flat sheet or
3 1 .3
5 .6 7
5 .6 7
2 .8 6
5 .8 1
1 6 .3
3 1 .3
1 6 .3
from an entirely different radius.
2 70
3 1 .3
3 .7 5
5 .8 9
1 6 .3
3 .7 5
The results of these tests are shown
3 1 .3
1 .8 8 5
2 .1 3
5 .9 6
1 6 .3
5 .6 7
in Pig. 15, in which the value of P
3 1 .3
5 .6 7
2 .0 9
3 .0 3
1 5 .7
5 .6 7
3 1 .3
2 .1 3
3 .0 9
3 .7 5
1 5 .7
obtained is plotted against the
3 1 .3
3 .7 5
3 .0 0
1 5 .7
3 .7 5
3 1 .3
1 .8 8 5
2 .7 0
2 .9 7
1 5 .7
5 .6 7
difference between the final and
3 1 .3
5 .6 7
2 .1 3
2 .7 4
1 5 .7
3 .7 5
initial curvatures. I t will be seen
5 .6 7
2 .0 5
1 .9 9 • 3 1 .3
1 5 .7
5 .6 7
3 1 .3
3 .7 5
2 .0 9
1 .9 9
1 5 .7
3 .7 5
th at the specimens with the high­
3 1 .3
3 .7 5
2 .0 9
1 .9 9
1 5 .7
1 .8 8 5
5 .6 9
6 .0 3
1 6 .3
est initial stresses, due to the
5 .6 9
5 .8 5
1 6 .3
springing, consistently gave the
5 .6 9
5 .8 5
1 6 .3
3 .7 5
5 .9 6
1 6 .3
lowest values of P. To be sure, th variation of P is within
5 .6 7
1 .8 9
1 5 .7
3 .7 5
1 5 .7
the limits of the scattering observed l other tests, so th at this
5 .6 7
5 .9 3
1 6 .3
variation could possibly be accidental This seems hardly likely,
1 .8 8 5
5 .9 5
1 6 .3
however, in view of the number of the tests and the fact that
two specimens of each kind were tested and the same tendency
O N F I N A L C O M P R E S S IV E S T R E N G T H 13
was exhibited by both specimens. The fact th a t all specimens
O riginal rad iu s of
O riginal rad iu s of
cu rv a tu re of
F a ilin g
F a ilin g
cu rv a tu re of
were cut from the same roll of material probably reduced the
sh e et, in.
lo a d , lb
sh e et, in .
lo a d ,lb
real scattering in this case.
— 0 .5 6 °
1 .5
— 0 .5 6 °
1 .5
These results have an important practical implication which is
— 4 . 3°
0 .5 6
quite obvious. They also bear out the contention th a t final
— 5 . 0“
0 .5 6
failure is precipitated by yielding of the material, as obviously
A ll cy lin d ers w ere of b rass, w ith d ia m eter 3 .7 5 in ., le n g th 6 .0 in ., th ic k ­
ness 0.00295 in ., a n d E 1 5 ,7 0 0 ,0 0 0 lb per sq in .
the initial stresses due to springing the sheet (present on the
° N e g a tiv e sig n in d ica te s original cu rvatu re w as in o p p o site d irectio n
outer and inner surfaces all over the cylinder) will combine with
from final cu rvatu re.
the stress due to other causes to produce yielding before it would
otherwise occur. Of course, if the wall is bent to the proper
where V2 = — +
V4 signifies the application of V 2twice;
shape before making the joint, some initial stresses will also be
present, but they will be much smaller and more localized than
and V8 four times. This equation is satisfied if
where the sheet is sprung to shape. Also, the cold-working of the
material may raise its yield point slightly, which would also fall
in with our contention.
Appendix 1. Theory on the Assumption of
Perfect Initial Shape and Infinitesimal
In a previous paper12 the author has shown th a t the conditions
for equilibrium of an element of the wall of a thin cylinder, under
uniform axial compression, when the displacement consists of
several waves around the circumference, can be simplified to
We neglect edge conditions entirely because, due to the small size
of the waves, it is not im portant whether there are an even num­
ber of waves in the circumference, or what the conditions are at
the ends of the cylinders. (In the tests, buckling frequently oc­
curred over only a part of the length or circumference.) Substi­
tuting [7] in [6] and using the symbols of the present paper, we
Equation [8] gives the values of P required to maintain various
states of equilibrium involving different values of X and S.
! N.A.C.A. R eport No. 479, pp. 13-15.
19<jy for th e s teel u sed
w as arou n d 5 7 ,0 0 0 lb per sq in . in all te s ts .
for th e b rass w as b etw een 2 8 ,0 0 0 an d 3 0 ,0 0 0 lb p er sq in . in all te s ts .
Buckling will take place as soon as P rises to the lowest of these
values. B y inspection, or using minimum principles, the lowest
-value of P obtainable from [8] is 2, obtained when the quantity
>(X + S ) 2/ X = 1. We thus obtain Equations [1] and [2], which
have been discussed.
The equations of equilibrium of an element in the x and »
directions can be taken the same as in the small-deflection
theory,12 because the only new internal forces which we are con­
sidering (which are not considered in the small-deflection theory)
are the large-deflection stresses, which form a part of Tx, T„ and
Tx,. And T x, T», and Tx, are fully considered in these equations:
Appendix 2. Theory Considering Initial Dis­
placements and Finite Deflections
T h e strains of the middle surface of the cylinder wall are ob-tained in terms of the displacements u, v, wlt w2 from the geomet­
rical relationships between them. We find, for the linear strains
in the x and s directions, the shear strain, the changes in curva­
tures in the x and s directions, and the unit twist
These equations will be satisfied if we take
The terms involving squares or products of derivatives are largedeflection strains representing the change in length of elements
due to their slope. The other terms are the same as in the smalldeflection theory.11 The second expressions for ex, kx, etc. are
obtained by substituting the relations
where / is the usual stress function, or Airy function,
except for the constant factor Et*/c, which is used to sim­
plify the results.
Equating the expressions for T*, T,, and T*, in [12] and
in [15] and solving for Cxj
and ta, we find
We next eliminate u and v between the three equations of [9]
by applying the operator — to the first equation, — to the
’ dx3
second equation, and subtracting these two equations from the
third equation, to which the operator ----- has been applied.
This gives us
in the first expressions, remembering that K ( = 1 + 2 — by defini­
tion) is a constant with respect to x and s, since w and wi are
assumed to have the same geometrical form.
The relations between the strains and the internal stresses,
given by Hooke’s law, are the same as used in the small-deflection
theory.12 The internal forces and moments per unit length of
section (see Fie. 11) are
Substituting [16 ] in this, we obtain the following relation betweei
the stress fu n ctio n /a n d the radial movement w
An expression analogous to [18], for the case of a flat plate without
initial displacement (r = <o,K = 1) was first obtained by von
K&nnfln.14 (However, the author made this derivation indepen­
dently, before learning of von Kdrmdn’s solution.)
The internal elastic energy is
[substituting Equations [15], [16], [13], and [10] in Equation
[19] we obtain an expression for the internal elastic energy in
terms of / and w. If / and w are harmonic functions of x and s,
this simplifies to
14 Enzyklopadie
der Math. Wiss., vol. 4, art. 27.
Equations [18] and [20] are general formulas which can be
used in solving many other large-deflection problems in which the
initial displacement can be taken as geometrically similar to the
final displacement, or as zero (in which case K = 1). For flat
sheets the first term on the right-hand side of [18] drops out. If
an approximate expression for w is assumed, an expression for /
can be derived from [18] and the boundary or other conditions,
after which [20] can be used to apply the principle of virtual work.
We assume for w the shape discussed before
F i g . 11
and take Wi — — w, as implied in the definition of K . SubstitutW
ing [21] in [18] and using the symbols X and S, we find
Fio. 12
From our knowledge of the physics of the problem we know
that Tx, T,, and Tx, will be harmonic functions, except for a
constant component of T x equal to — t<r. Hence, from [22], f
can be taken as
The coefficients of the terms in [23] were found by taking them
as unknowns, substituting [23] in [22], and solving for the values
of the coefficients satisfying [22 ] for any values of x or s. The co­
efficient of the second term in [21] was determined in a similar
way, so as to satisfy the condition that the part of T , (found by
using [23] in [15]) independent of s shall vanish, as discussed in
the first part of the paper. The constant C in [23] is found from
the condition that the constant part of Tx shall equal — to- or
Using [21], [23], and [24] in [20] and integrating over the
circumference and length, we find the internal elastic energy
F iq .
R e s u l t s o f T e s t s o p C y l in d e r s W i t h
I n i t ia l S t r e s s e s D u e t o F o r m in g
D if f e r e n t
these operations are carried out, after substituting [21 ] in [26] and
remembering that K is a function of W, we obtain an expression
for P in terms of W, Wi, X , and S
where h is the length of the cylinder, or of the part of the cylinder
considered. The last term in [25] evidently represents the elas­
tic work due to the ordinary elastic shortening of the cylinder
under the load. This has no effect in our problem and the deriva­
tion could have been simplified by omitting the non-harmonic
part of [23 ] which produces this term. However, the justification
for such an omission might not have been clear.
The work done by the external forces during a virtual displace­
ment dW is equal to 2irrta times the average distance the cylin­
der is shortened during such a displacement, or
B y the principle of virtual work, this can be equated to the
change of E. during the displacement dW which is — dW.
It will be observed that if W i is set equal to zero and terms con­
taining W 2 are neglected as second-order terms, [27] reduces to
Equation [8] of the classical theory.
In carrying out the integrations of [25] and [26] it is assumed
that L x and L , are even multiples of the length and the circum­
ference. This involves little error because of the small size of L x
and L„ as discussed in the main part of this paper.
We shall now set up the condition for yielding of the material
at any point. According to the maximum-shear-energy theory16
which is generally considered to be the most exact expression of
11 See “Plasticity,” by A. Nadai, McGraw-Hill, N. Y., 1931.
the condition for yielding, plastic flow under combined stresses
at any point commences when
calculations indicate that the material at the surface of the
cylinder wall in the nodes of the wave form is in about as unfavor­
able a state as any, in most cases, at least. If we take x = s =
0 and z = t/2 in this expression we obtain
where <ri, o-2, and <r3 are the principal stresses at the point and
ay is the yield-point stress in simple tension. In our case the
stress in the radial direction can be taken as zero and as one of
the principal stresses, while the other two principal stresses, in
the plane of the cylinder wall, are given by the formula
where <Txj
and 0*8 are the normal and shear stresses on planes
Equation [4], discussed in the first part of the paper, can now
be combined with Equations [27] and [32] to eliminate W and
Wi, and obtain P as a function of P y, fi, X, S, n, and a. The
influence of n, which enters [32], is not important and a value of
0.3 can be taken for it for all engineering metals. We shall also
assume that X = S because of the difficulty of checking this ex­
perimentally verified relation, as previously discussed. With
these assumptions [27] and [32] are somewhat simplified
0.7 0.9
and Equation [4] becomes
Combining [27'], [32'], and [4'], we can obtain P as a function of
P y, r/t, X , n, and a. Then we can determine n so as to make the
value of X , at which P is a minimum, coincide with test results,
and determine a to bring the general magnitude of the values of
P down to the level of test results.
The complexity of the equations made it impractical to do this
directly. Actually, various values of W, Wi, and X were as­
sumed, which enabled the corresponding values of P and P y to be
found from [27'] and [32']. This gave sufficient data to plot
F ig. 16
T h e o r e t i c a l R e l a t i o n B e t w e e n P , P y, Wi, a n d
f o b X = S — 0.07
perpendicular to the x and s directions.
for the principal ^tresses in [28], we find
Using these values
The values of ax, <rs, and <rXs at a point in the cylinder wall a dis­
tance z from the middle plane are, assuming a linear distribution
of stress,
Substituting [31] in [30] and using the expressions for TX, G X
etc. [15], [13], [10], and finally [21], [23], and [24], we obtain an
expression for P in terms of P„, W, Wi, X , S, x, s, and z. B y
using minimum theory we could determine the value of x, s,
&nd z at which P is a minimum—that is, the point at which yield­
ing first occurs, at the lowest value of a. But the expression is
too complex to make this very practical. Trials and elementary
families of curves giving the relation of P and Pt
with W and W 1 for several values of X . Fig. 16 shows such families
of curves for X = 0.07, and similar families were drawn for X =
0.04 and X = 0.10. Then, assuming values of a, n, X, and r/t,
we find the value Wi from [4']. Taking E / c < t u = 165, as in the
tests made by the author, P„ can be calculated, and the corre­
sponding values of W and P found from the curves such as shown
in Fig. 16. Then by comparing the different results obtained for
the different values of X , and plotting P against X for the same
values of r/t, we can roughly determine the value of X at which
P is a minimum. This gives sufficient data to plot curves such
as shown in Figs. 4 and 5. This process was repeated with differ­
ent values of a and n until the combination used in plotting Figs.
4 and 5 was found.
A cknow ledgm ent
The author wishes to acknowledge the assistance of Messrs.
L. Secretan and K. W. Donnell, who carried out most of the ex­
periments, and he wishes to thank E. L. Shaw and Sylvia K.
Donnell for help in preparing this paper.